SPINORS IN THE DYNAMICAL THEORY OF SPINNING PARTICLES

1952 ◽  
Vol 30 (3) ◽  
pp. 226-234 ◽  
Author(s):  
S. Shanmugadhasan
Keyword(s):  
Hamiltonian Dynamics ◽  
Equations Of Motion ◽  
Dynamical Theory ◽  
Total Spin ◽  
System Of Equations ◽  
Poisson Brackets ◽  
Spin Angular Momentum ◽  
Canonical Variables ◽  
Spinor Form ◽  
Set Up

The correspondence between self-dual six-vectors and symmetric spinors of the second rank is used to put into spinor form the rotational equations of motion of a particle analogous to a pure gyroscope or to a symmetrical top. These equations are then split up into an equivalent system of equations in terms of spinors of the first rank. The Lagrangian of each system is set up, and the canonically conjugate variables obtained from it in terms of covariant spinors. But the canonical variables, being not all independent, lead to weak equations in the sense of Dirac. Therefore, Dirac's generalized Hamiltonian dynamics is used in the canonical formulation in terms of Poisson Brackets. The detailed discussion of the symmetrical top case shows that, though the fundamental Poisson Brackets for the total spin angular momentum and the "spin" are the usual ones, those Poisson Brackets-involving the derivative of the "spin" are not unique.

THE QUANTIZATION OF CLASSICAL SPIN THEORY

1953 ◽  
Vol 31 (1) ◽  
pp. 1-10 ◽  
Author(s):  
S. Shanmugadhasan
Keyword(s):  
Hamiltonian Dynamics ◽  
Equations Of Motion ◽  
Supplementary Condition ◽  
Poisson Brackets ◽  
Rest System ◽  
Spin Tensor ◽  
Constraint Condition ◽  
Set Up ◽  
Jacobi Equations ◽  
Quantum Formulation

The antisymmetric spin tensor of rank two used to describe the rotational motion of a particle is assumed to satisfy the constraint condition that the velocity 4-vector is orthogonal to it. Since the dipole moment is proportional to the spin tensor, this condition leads always to a purely magnetic dipole in the rest system of the particle. Frenkel has indicated how the action principle for the classical equations of motion can be set up treating the above constraint condition as a supplementary condition. The Hamiltonian dynamics of a system having supplementary conditions and Lagrange undetermined multipliers has been discussed recently by Dirac. Dirac's method and results previously obtained give the required Hamilton–Jacobi equations and Poisson Brackets of the dynamical variables. The cases where the particle behaves like a pure gyroscope and a symmetrical top are treated. When there is an interacting field, it is assumed that the action of the Held is given by the effective 4-vector potential without further specification. The orthogonality of the velocity to the tensor dual to the spin tensor can be imposed as an alternative constraint condition. This possibility is discussed briefly. The quantum formulation is completed with the help of the standard analogy rules.

THE DYNAMICAL THEORY OF MAGNETIC MONOPOLES

1952 ◽  
Vol 30 (3) ◽  
pp. 218-225 ◽  
Author(s):  
S. Shanmugadhasan
Keyword(s):  
Electromagnetic Field ◽  
Equations Of Motion ◽  
Natural Generalization ◽  
Magnetic Monopoles ◽  
Dynamical Theory ◽  
The Other ◽  
Field Tensor ◽  
Physical Content ◽  
Electromagnetic Field Tensor ◽  
Set Up

The theory of electric charges and magnetic monopoles has been set up by Dirac by expressing the electromagnetic field tensor in terms of one four-potential and of the variables describing the strings attached to each magnetic mono-pole. In this reformulation of Dirac's theory the field tensor is expressed in terms of two four-potentials, one corresponding to charges and the other to monopoles, and the action principle for the equations of motion is set up in terms of the two four-potentials and of the tensors dual to them. Thus there is formal symmetry as far as is possible in the treatment of the charges and the monopoles. Also the mathematics is direct and neat. Though the physical content is the same as that of Dirac, a natural generalization of the Fermi form of electrodynamics subject to the restriction that the same particle cannot have both charge and monopole is obtained here.

THE QUANTIZATION OF THE CLASSICAL THEORY OF SPINNING PARTICLES

1951 ◽  
Vol 29 (6) ◽  
pp. 593-612 ◽  
Author(s):  
S. Shanmugadhasan
Keyword(s):  
Dipole Moment ◽  
Classical Theory ◽  
Jacobi Equation ◽  
Rest Mass ◽  
Total Spin ◽  
Poisson Brackets ◽  
Spin Angular Momentum ◽  
Scalar Products ◽  
Kinetic And Potential Energies ◽  
Hamilton Jacobi Equation

The classical theory of particles, possessing charge and dipole moment, and moving in an electromagnetic field, is considered on the assumptions that there is no constraint connection between the rotational variables and the velocity of the particle, and that the two invariant squares of the dipole moment six–vector are constants of the motion. Two different schemes are obtained according as the two invariant scalar products of the dipole moment and total spin angular momentum six–vectors are or are not constants of the motion. The Bhabha–Corben theory fits into the former scheme. The classical schemes are put into canonical form by using for each particle the relativistic connection between the momenta and the rest-mass, modified to include the effect of the kinetic and potential energies due to spin and dipole moment, as the Hamilton–Jacobi equation and the usual Poisson brackets for the translational and total spin variables. The Wentzel field and the λ-limiting process are used mainly in dealing with the field. The variational principle for the Bhabha–Corben equations is given with the field treated according to the limiting process of Dirac or the relativistic cutoff method of Feynman. The quantization is completed by using the analogy rules. The changes required when the interacting field is a vector meson field are discussed.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

scholarly journals Shallow water waves in a rotating rectangular basin

2000 ◽  
Vol 24 (10) ◽  
pp. 649-661 ◽  
Author(s):  
Mohamed Atef Helal
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Equations Of Motion ◽  
System Of Nonlinear Equations ◽  
System Of Equations ◽  
Order Of Approximation ◽  
Shallow Water Waves ◽  
Nonlinear System Of Equations ◽  
Rectangular Basin ◽  
Shallow Water Approximation

This paper is mainly concerned with the motion of an incompressible fluid in a slowly rotating rectangular basin. The equations of motion of such a problem with its boundary conditions are reduced to a system of nonlinear equations, which is to be solved by applying the shallow water approximation theory. Each unknown of the problem is expanded asymptotically in terms of the small parameterϵwhich generally depends on some intrinsic quantities of the problem of study. For each order of approximation, the nonlinear system of equations is presented successively. It is worthy to note that such a study has useful applications in the oceanography.

scholarly journals On the Dirac eigenvalues as observables of the on-shell N = 2D = 4 Euclidean supergravity

Open Physics ◽  
2008 ◽  
Vol 6 (4) ◽  
Author(s):  
Ion Vancea
Keyword(s):  
Phase Space ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Gauge Transformations

AbstractWe generalize previous works on the Dirac eigenvalues as dynamical variables of Euclidean gravity and N =1 D = 4 supergravity to on-shell N = 2 D = 4 Euclidean supergravity. The covariant phase space of the theory is defined as the space of the solutions of the equations of motion modulo the on-shell gauge transformations. In this space we define the Poisson brackets and compute their value for the Dirac eigenvalues.

scholarly journals IV. On the dynamical theory of incompressible viscous fluids and the determination of the criterion

1895 ◽  
Vol 186 ◽  
pp. 123-164 ◽  
Keyword(s):  
Singular Solution ◽  
Physical State ◽  
Theoretical Calculations ◽  
Equations Of Motion ◽  
Dynamical Theory ◽  
Viscous Fluids ◽  
Linear Functions ◽  
Incompressible Viscous Fluids ◽  
Theoretical Results

1. The equations of motion of viscous fluid (obtained by grafting on certain terms to the abstract equations of the Eulerian form so as to adapt these equations to the case of fluids subject to stresses depending in some hypothetical manner on the rates of distortion, which equations Navier seems to have first introduced in 1822, and which were much studied by Cauchy and Poisson) were finally shown by St. Venant and Sir Gabriel Stokes, in 1845, to involve no other assumption than that the stresses, other than that of pressure uniform in all directions, are linear functions of the rates of distortion, with a co-efficient depending on the physical state of the fluid. By obtaining a singular solution of these equations as applied to the case of pendulums in steady periodic motion, Sir G. Stokes was able to compare the theoretical results with the numerous experiments that had been recorded, with the result that the theoretical calculations agreed so closely with the experimental determinations as seemingly to prove the truth of the assumption involved. This was also the result of comparing the flow of water through uniform tubes with the flow calculated from a singular solution of the equations so long as the tubes were small and the velocities slow. On the other hand, these results, both theoretical and practical, were directly at variance with common experience as to the resistance encountered by larger bodies moving with higher velocities through water, or by water moving with greater velocities through larger tubes. This discrepancy Sir G. Stokes considered as probably resulting from eddies which rendered the actual motion other than that to which the singular solution referred and not as disproving the assumption.

Differential-Geometric Methods in Multibody Dynamics and Control

2005 ◽  
Author(s):  
P. Maißer
Keyword(s):  
Configuration Space ◽  
High Performance ◽  
Multibody System ◽  
Equations Of Motion ◽  
Geometric Approach ◽  
Motion Equations ◽  
Constrained Mechanical Systems ◽  
Dynamics And Control ◽  
Set Up ◽  
Lagrangian Motion

This paper presents a differential-geometric approach to the multibody system dynamics regarded as a point dynamics in a n-dimensional configuration space Rn. This configuration space becomes a Riemannian space Vn the metric of which is defined by the kinetic energy of the multibody system (MBS). Hence, all concepts and statements of the Riemannian geometry can be used to study the dynamics of MBS. One of the key points is to set up the non-linear Lagrangian motion equations of tree-like MBS as well as of constrained mechanical systems, the perturbed equations of motion, and the motion equations of hybrid MBS in a derivative-free manner. Based on this approach transformation properties can be investigated for application in real-time simulation, control theory, Hamilton mechanics, the construction of first integrals, stability etc. Finally, a general Lyapunov-stable force control law for underactuated systems is given that demonstrates the power of the approach in high-performance sports applications.

scholarly journals A Study of the Stability of a Plate-Like Load Towed beneath a Helicopter

1971 ◽  
Vol 13 (5) ◽  
pp. 330-343 ◽  
Author(s):  
D. F. Sheldon
Keyword(s):  
Equations Of Motion ◽  
Physical Parameters ◽  
Recent Experience ◽  
Wind Tunnel Testing ◽  
Stability Derivatives ◽  
Direct Indication ◽  
Metric Dimensions ◽  
Set Up ◽  
The Stability ◽  
Derivative Information

Recent experience has shown that a plate-like load suspended beneath a helicopter moving in horizontal forward flight has unstable characteristics at both low and high forward speeds. These findings have prompted a theoretical analysis to determine the longitudinal and lateral dynamic stability of a suspended pallet. Only the longitudinal stability is considered here. Although it is strictly a non-linear problem, the usual assumptions have been made to obtain linearized equations of motion. The aerodynamic derivative data required for these equations have been obtained, where possible, for the appropriate ranges of Reynolds and Strouhal number by means of static and dynamic wind tunnel testing. The resulting stability equations (with full aerodynamic derivative information) have been set up and solved, on a digital computer, to give direct indication of a stable or unstable system for a combination of physical parameters. These results have indicated a longitudinal unstable mode for all practical forward speeds. Simultaneously the important stability derivatives were found for this instability and modifications were made subsequently in the suspension system to eliminate the instabilities in the longitudinal sense. Throughout this paper, all metric dimensions are given approximately.

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